Four-fermion interactions and scale invariance

Four-fermion interactions of the current-current type with U(n) symmetry, in one space and one time dimension, are investigated. It is shown that the equations of motion yield scale-invariant solutions only for two values of the coupling gᵥ of the SU(n) currents, namely gᵥ=0 and gᵥ=4π(n+1). This holds for any value of the coupling gB of the U(1) currents. For the above two values of gᵥ and any gB the theory is solved completely. Operator products of spinor fields are shown to be equal to c-number functions singular on the light cone times analytic bilocal operators expressed in terms of currents and free spinor fields. The currents are free for the above two values of gᵥ. The connection with the coupling as defined through four-point functions is discussed, and it turns out that the combination corresponding to SU(n) coupling is zero for both solutions. However, the solution for gᵥ=4π(n+1) exhibits nontrivial four-point functions also for gB=0. It is shown, in an expansion around gᵥ=0, that there is only one Callan-Symanzik function β which depends only on gᵥ and that gᵥ=0 is relevant to the ultraviolet limit of the gᵥ>0 theories. When mass terms are introduced, this still holds in an infinite interval for gB, which is bounded below by a certain negative value and in which the mass term is soft.